# The Beltrami Equation

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*Tadeusz Iwaniec; Gaven Martin*

The “measurable Riemann Mapping
Theorem” (or the existence theorem for quasiconformal mappings)
has found a central rôle in a diverse variety of areas such as
holomorphic dynamics, Teichmüller theory, low dimensional
topology and geometry, and the planar theory of PDEs. Anticipating
the needs of future researchers, the authors give an account of the
“state of the art” as it pertains to this theorem, that is,
to the existence and uniqueness theory of the planar Beltrami
equation, and various properties of the solutions to this equation.
The classical theory concerns itself with the uniformly elliptic case
(quasiconformal mappings). Here the authors develop the theory in the
more general framework of mappings of finite distortion and the
associated degenerate elliptic equations.

The authors recount aspects of this classical theory for the
uninitiated, and then develop the more general theory. Much of this
is either new at the time of writing, or provides a new approach and
new insights into the theory. Indeed, it is the substantial recent
advances in non-linear harmonic analysis, Sobolev theory and geometric
function theory that motivated their approach here. The concept of a
principal solution and its fundamental role in understanding
the natural domain of definition of a given Beltrami operator is
emphasized in their investigations. The authors believe their results
shed considerable new light on the theory of planar quasiconformal
mappings and have the potential for wide applications, some of which
they discuss.

#### Table of Contents

# Table of Contents

## The Beltrami Equation

- Contents vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. Quasiconformal Mappings 516 free
- Chapter 3. Partial Differential Equations 1122
- Chapter 4. Mappings of Finite Distortion 1728
- Chapter 5. Hardy Spaces and BMO 2738
- Chapter 6. The Principal Solution 3344
- Chapter 7. Solutions for Integrable Distortion 3950
- 7.1. Distortion in the Exponential Class 4152
- 7.2. An Example 4253
- 7.3. Results 4354
- 7.4. Distortion in the Subexponential Class 4556
- 7.5. An Example 4556
- 7.6. Further Generalities 4758
- 7.7. Existence Theory 4859
- 7.8. Global Solutions 6071
- 7.9. Holomorphic Dependence 6475
- 7.10. Examples and Non-Uniqueness 6778
- 7.11. Equations in the Plane 7384
- 7.12. Compactness 7788
- 7.13. Removable Singularities 7990
- 7.14. Final Comments 8091

- Chapter 8. Some Technical Results 8192
- Bibliography 89100